On Dual Bade Theorem in Locally Convex C(k)-Modules

Gök, Ömer

doi:10.1515/dema-1999-0416

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“…Let N be a weak* closed unital subalgebra of (m*{Bi) ) and suppose that T G L(X') leaves invariant all weak * closed TV-invariant subspaces of X'. The dual reflexivity theorem [5,Corollary 1] …”

Section: Theorem 4 Let X Be a Barrelled Locally Convex C(k)-module mentioning

confidence: 99%

“…where closure is taken with respect to the weak * operator topology, by corollary 1, [5]. We say that an equicontinuous Boolean algebras B of projections in the locally convex space X is strongly equicontinuous if {E n } converges to zero in L(X) whenever {E n } C B is a disjoint sequence.…”

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confidence: 99%

“…We say that an equicontinuous Boolean algebras B of projections in the locally convex space X is strongly equicontinuous if {E n } converges to zero in L(X) whenever {E n } C B is a disjoint sequence. We note that rn*(B\) is strongly equicontinuous in L(X'), [5]. Let A be a collection of continuous linear operators on X.…”

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On Reflexivity of Closed Unital Subalgebras in Locally Convex Spaces

Gök¹

2001

Demonstratio Mathematica

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IntroductionIn this paper, we are interested in Bade's reflexivity theorem indicated in [2] in barrelled locally convex Hausdorff spaces. The Bade theorem says that a continuous linear operator T on a Banach space X belongs to the strongly closed algebra generated by a a-complete Boolean algebra B of projections if and only if T leaves invariant each B-invariant subspace of X. As a consequence, it is shown that each unital closed subalgebra of the span closure of an equicontinuous Boolean algebra of projections in the barrelled spaces is reflective. It extends Proposition 5.8 in [2].For the concepts and notations not explained in this paper we refer to the books [1], [7]. Throughout, X will denote a quasicomplete locally convex Hausdorff topological vector space. By L(X) we denote the space of continuous linear operators from X into X. The identity operator in X is denoted by I,X' will denote the continuous dual of X. The adjoint of an operator T G L(X) is denoted by T". Let B be an equicontinuous Boolean algebra of projections in the barrelled locally convex Hausdorff space X. It is known that if X is a barrelled space, then X' is cr(X', X)-quasicomplete, [7, page 148]. Let p be the family of all continuous semi-norms in X, then the topology in X is equal to the topology generated by p.

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Section: Theorem 4 Let X Be a Barrelled Locally Convex C(k)-module mentioning

confidence: 99%

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confidence: 99%